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Hemispherical Power Asymmetry
郭宗宽
昆明2013.11.2
anomalies in CMB map
• the quadrupole-octopole alignment• power deficit at low-l • hemispherical asymmetry• parity asymmetry• the cold spot• ……
∑𝑚
𝑚2|𝑎𝑙𝑚(�̂�)|2
∆𝑇 (�̂�)𝑇
=∑𝑙 𝑚
𝑎𝑙𝑚𝑌 𝑙𝑚(�̂�)
𝑎𝑙𝑚=∫𝑑�̂�𝑌 𝑙𝑚∗(�̂�)
∆𝑇 (�̂�)𝑇
⟨𝑎𝑙𝑚∗𝑎𝑙′ 𝑚′ ⟩=𝛿𝑙 𝑙′ 𝛿𝑚𝑚′ 𝐶𝑙
𝐶 (𝜃 )=⟨ ∆𝑇 (�̂�1)𝑇
∆𝑇 (�̂�2)𝑇 ⟩
¿14 𝜋∑
𝑙
(2𝑙+1)𝐶𝑙𝑃 𝑙 (cos𝜃)
𝑃+¿ (𝑙 )=∑
𝑛=2
𝑙
cos2 (𝑛 𝜋2 )𝑛(𝑛+1 )2 𝜋
𝐶𝑛 ¿
𝑃− (𝑙 )=∑𝑛=2
𝑙
sin2(𝑛𝜋2 )𝑛(𝑛+1)2𝜋
𝐶𝑛
𝑔 (𝑙 )= 𝑃+¿(𝑙)
𝑃−( 𝑙)¿
data analysis
d (�̂�)= (1+ 𝐴�̂� ∙ �̂�) s (�̂�)+n (�̂�)
the CMB temperature sky maps is modeled as
the likelihood is given by
−2ℒ ( 𝐴 , �̂� ,𝑞 ,𝑛)=d𝑇C−1d+ log|C|
C.Gordon, W.Hu, D.Huterer and T.M.Crawford, arXiv:astro-ph/0509301
H.K.Eriksen, A.J.Banday, K.M.Gorski, F.K.Hansen and P.B.Lilje, arXiv:astro-ph/0701089
C (�̂� ,�̂�)=S (�̂� ,�̂� )+N+F
the full covariance matrix is
S (�̂� ,�̂� )=(1+𝐴�̂� ∙ �̂�)S iso (�̂� ,�̂� )(1+𝐴�̂� ∙�̂�)
Siso (�̂� ,�̂� )= 14𝜋 ∑
𝑙
(2 𝑙+1)𝐶𝑙𝑃 𝑙(�̂� ∙�̂�)
𝐶𝑙=𝑞 ( 𝑙𝑙0 )𝑛
𝐶𝑙❑fid
• WMAP3: (l, b) = (225◦,−27◦), A=0.114
• WMAP5(l<64, 4.5◦): (l, b) = (224◦,−22◦), A=0.0720.022
• Planck:J.Hoftuft, H.K.Eriksen, A.J.Banday, K.M.Gorski, F.K.Hansen and P.B.Lilje, arXiv:0903.1229
P.A.R.Ade et al. [Planck Collaboration], arXiv:1303.5083
preferred dipole directions dependence on smoothing scale
Comment: the dipole anisotropy induced by our velocity
𝑇 (�̂� )= 𝑇 ′ (�̂� ′)𝛾(1−�̂� ∙ 𝛽)
�̂�=�̂�′+[ (𝛾−1 )�̂�′ ∙ �̂�+𝛾𝛽 ] �̂�
𝛾 (1+�̂�′ ∙𝜷)
𝛿 𝐼𝜈 (𝜈 ,�̂�)𝑑 𝐼𝜈/𝑑𝑇
=𝑇0 �̂� ∙𝜷+𝛿𝑇 ′(�̂�−𝛻 (�̂� ∙𝜷))(1+𝑏𝜈 �̂� ∙𝜷)
the temperature and direction in the observed frame
the inferred temperature fluctuations
𝛽≡𝑣𝑐
=1.23×10− 3
(l, b) = (264◦,48◦)
models
• a parameterization [astro-ph/0509301]• our peculiar motion [arXiv:1304.3506]• a gauge field [arXiv:1302.7304]• anisotropic metric [arXiv:1303.6058]• a modulation of a cosmological parameter
[arXiv:1303.6949]• a superhorizon perturbation
[arXiv:0806.0377]• …… [arXiv:yymm.xxxx]
a superhorizon perturbation
A.L.Erickcek, M.Kamionkowski and S.M.Carroll, arXiv:0806.0377
𝛿𝑇𝑇
≃13
Φ=13ℛ (matter −dominated )the Sachs-Wolfe effect
𝜙(𝑡∗)⟶𝑘(𝑡∗)𝒫ℛ (𝑘)=(𝐻�̇� )2
( 𝐻2𝜋 )2
𝑘=𝑎𝐻
the diploe modulation of curvature perturbation
𝒫ℛ1/2 (𝑘 , 𝒙 )=(1+𝐴 �̂� ∙ 𝒙
𝑥 ls)𝒫ℛ
1/2 (𝑘 )
the asymmetry A is
𝐴=|𝛻𝒫ℛ
1/2|𝒫ℛ1 /2 𝑥 ls=𝑑 ( ln𝒫ℛ
1/2¿ ¿𝑑 ( ln𝑘¿
¿𝑑 ( ln𝑘¿ ¿𝑑𝑡
𝑑𝑡𝑑𝜙
|𝛻𝜙|𝑥 ls
¿ 12
(𝑛ℛ −1 )𝐻 (1−𝜖 ) 1�̇�
(𝑘¿¿𝐿𝛿𝜙𝐿)𝑥 ls=(1−𝜖)(𝑛ℛ −1 )
2(𝑘𝐿𝑥ls )(𝐻�̇� 𝛿𝜙𝐿)¿
¿ (1−𝜖)(𝑛ℛ −1 )
2(𝑘𝐿𝑥 ls)𝒫ℛ ,𝐿
1/2
a single modulated mode𝒫ℛ (𝑘)=𝒫ℛ ,𝐿𝛿(ln𝑘− ln𝑘𝐿)
D.H.Lyth, arXiv:1304.1270
Z.G.Liu, Z.K.Guo and Y.S.Piao, arXiv:1304.6527
𝐶𝑙=4𝜋∫0
∞𝑑𝑘𝑘
𝑇 𝑙2(𝑘)𝒫ℛ (𝑘)
the CMB angular power spectrum
the GZ effect (the Sachs-Wolfe approximation)
𝐶2GZ=4𝜋
25∫0
1/𝑥 ls 𝑑𝑘𝑘 ( 𝑘2 𝑥 ls
2
15 )2𝒫ℛ (𝑘 )=4𝜋
25 (𝑘𝐿2𝑥 ls
2
15 )2𝒫ℛ ,𝐿
√𝐶2GZ≲1.8×10− 5⟹(𝑘¿¿𝐿𝑥 ls)
4𝒫ℛ ,𝐿≲16×10− 8¿
⟹(𝑘𝐿𝑥 ls)𝒫ℛ ,𝐿
12≲ 0.02
observational constraint
• a single-field slow-roll inflation
• a curvaton-type field
• a bounce inflation
𝐴 𝒪(10¿¿−4)¿
𝐴=65𝑓 NL (𝑘𝐿𝑥 ls )𝒫ℛ ,𝐿
1 /2 , 𝑓 𝑁𝐿≳8
(𝑛ℛ −1 ) 3 , ϵ 3 , 𝐴 0.06
A.L.Erickcek, M.Kamionkowski and S.M.Carroll, arXiv:0806.0377
D.H.Lyth, arXiv:1304.1270
Z.G.Liu, Z.K.Guo and Y.S.Piao, arXiv:1304.6527
𝒫ℛ=𝒫ℛinf 2𝜋𝑘|𝐶1−𝐶2|
2
𝒫ℛinf=𝐴inf( 𝑘𝑘0
)𝑛inf −1
primordial power spectrum of curvature perturbations
• semispherical power asymmetry• power deficit on large angular scales• signals in the TE and EE spectra?• scale dependent? A<0.015 (99% CL)
outlook
Thanks!